. . . Ubuntu 14.04, installed on a . . . laptop . . . decided to use an alternative laptop . . . install the existing SSD . . . without wiping the OS . . .
. . . proprietary drivers . . . machines are . . . two different manufacturers . . .
I have pulled HDD's out of one system, and had them boot without error, in another. SSD's should be no different than HDD's.
It is possible that the Ubuntu will just up and run on the new notebook, however, there are some things that have to be checked first.
Start with the architecture. Are they both 64bit, or both 32bit, or is one 64 and the other 32?
You can run a 32bit OS on a 64bit system, but you will be limited by the OS as to what hardware features you can use.
You can not run a 64bit OS on a 32bit system.
Next, look at the optional hardware.
First of all, make sure that the hard drive controllers are compatible. If one is using IDE mode, and the other AHCI, you will run into problems with the bootstrap. Set the new computer's controller, in the BIOS, to match the old one. (I believe that SSD's have to run in AHCI, but I could be wrong.)
If the GPU's are different, you will have a problem. You should be able to overcome this obstacle by un-installing any proprietary video drivers you have. Ubuntu should re-detect the new GPU. You may have to re-set your desktop resolution, though.
The other obstacles are sound and network interface. These two should not prevent Linux from booting, though.
Linux distro's have become much better than Windows at detecting hardware changes and compensating for them.
That is why live CD/USB disks work so well with almost any hardware. I have not yet found a system that I can't plug my 32bit Xubuntu USB stick into and have it work.
I suggest you try it, and see. If it boots, then you are laughing. If it won't boot, then you will have no choice but to re-install. You won't know, until you try.
By the way, there are ways to re-install Ubuntu, without loosing you personal data. We can cross that bridge, when we come to it!